In
research I strive to develop fundamental computational models for
physical simulation, computer animation, and geometric modeling. I draw
on ideas from graphics, applied mathematics, differential geometry, and
engineering.

I am currently teaching Discrete Mathematics, as well as Advanced Topics in Physical Simulation and Geometric Modeling. If you are on-campus, you can access the Wiki page: . I also teach Computer Animation, Physical Simulation, and Discrete Differential Geometry.

Discrete Differential Geometry

ddg.cs.columbia.edu offers a glimpse of an exciting emerging field. Much of the work you see below is built on the foundations laid out by DDG.

Abstract:We extend the Asynchronous Contact Mechanics algorithm and improve its performance by two orders of magnitude, using only optimizations that do not compromise ACM’s three guarantees of safety, progress, and correctness. The key to this speedup is replacing ACM’s timid, forward-looking mechanism for detecting collisions—locating and rescheduling separating plane kinetic data structures—with an optimistic speculative method inspired by Mirtich’s rigid body Time Warp algorithm [2000]. Time warp allows us to perform collision detection over a window of time containing many of ACM’s asynchronous trajectory changes; in this way we cull away large intervals as being collision free. Moreover, by replacing force processing intermingled with KDS rescheduling by windows of pure processing followed by collision detection, we tranform an algorithm that is very difficult to parallelize into one that is embarrassingly parallel. [PDF] [bib] [Project] [Video]

Abstract:Resolving simultaneous impacts is an open and significant problem in collision response modeling. Existing algorithms in this domain fail to fulfill at least one of five physical desiderata. To address this we present a simple generalized impact model motivated by both the successes and pitfalls of two popular approaches: pair-wise propagation and linear complementarity models. Our algorithm is the first to satisfy all identified desiderata, including simultaneously guaranteeing symmetry preservation, kinetic energy conservation, and allowing break-away. Furthermore, we address the associated problem of inelastic collapse, proposing a complementary generalized restitution model that eliminates this source of nontermination. We then consider the application of our models to the synchronous time-integration of large-scale assemblies of impacting rigid bodies. To enable such simulations we formulate a consistent frictional impact model that continues to satisfy the desiderata. Finally, we validate our proposed algorithm by correctly capturing the observed characteristics of physical experiments including the phenomenon of extended patterns in vertically oscillated granular materials.
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Abstract:We present an interactive animation editor for complex deformable object animations. Given an existing animation, the artist directly manipulates the deformable body at any time frame, and the sur- rounding animation immediately adjusts in response. The auto- matic adjustments are designed to respect physics, preserve detail in both the input motion and geometry, respect prescribed bilateral contact constraints, and controllably and smoothly decay in space- time. While the utility of interactive editing for rigid body and ar- ticulated figure animations is widely recognized, a corresponding approach to deformable bodies has not been technically feasible be- fore. We achieve interactive rates by combining spacetime model reduction, rotation-strain coordinate warping, linearized elasticity, and direct manipulation. This direct editing tool can serve the final stages of animation production, which often call for detailed, direct adjustments that are otherwise tedious to realize by re-simulation or frame-by-frame editing. [PDF] [bib] [Project] [Video]

Abstract:We present the first reduced-dimensional technique to simulate the dynamics of thin sheets of viscous incompressible liquid in three dimensions. Beginning from a discrete Lagrangian model for elastic thin shells, we apply the Stokes-Rayleigh analogy to derive a simple yet consistent model for viscous forces. We incorporate nonlinear surface tension forces with a formulation based on minimizing discrete surface area, and preserve the quality of triangular mesh elements through local remeshing operations. Simultaneously, we track and evolve the thickness of each triangle to exactly conserve liquid volume. This approach enables the simulation of extremely thin sheets of viscous liquids, which are difficult to animate with existing volumetric approaches. We demonstrate our method with examples of several characteristic viscous sheet behaviors, including stretching, buckling, sagging, and wrinkling. [PDF] [bib] [Project] [Video]

Abstract:It is well established that in certain domains, noisy inputs can be reliably combined to obtain a better answer than any individual. It is now possible to consider the crowdsourcing of physical actions, commonly used for creative expressions such as drawing, shading, and singing. We provide algorithms for converting low-quality input obtained from the physical actions of a crowd into high-quality output. The inputs take the form of line drawings, shaded images, and songs. We investigate single-individual crowds (multiple inputs from a single human) and multiple-individual crowds. [PDF] [bib] [Project]

Abstract:An asynchronous, variational method for simulating elastica in complex contact and impact scenarios is developed. Asynchronous Variational Integrators (AVIs) are extended to handle contact forces by associating different time steps to forces instead of to spatial elements. By discretizing a barrier potential by an infinite sum of nested quadratic potentials, these extended AVIs are used to resolve contact while obeying momentum- and energy-conservation laws. A series of two- and three-dimensional examples illustrate the robustness and good energy behavior of the method. [PDF] [bib] [Project]

Abstract:We propose an example-based approach for simulating complex elastic material behavior. Supplied with a few poses that characterize a given object, our system starts by constructing a space of prefered deformations by means of interpolation. During simulation, this example manifold then acts as an additional elastic attractor that guides the object towards its space of prefered shapes. Added on top of existing solid simulation codes, this example potential effectively allows us to implement inhomogeneous and anisotropic materials in a direct and intuitive way. Due to its example-based interface, our method promotes an art-directed approach to solid simulation, which we exemplify on a set of practical examples.
[PDF] [bib] [Project] [Video]

Abstract:We present a continuum-based discrete model for thin threads of viscous fluid based on the Rayleigh analogy to elastic rods, demonstrating canonical coiling, folding, and breakup in dynamic simulations. Our derivation emphasizes space-time symmetry, which ultimately sheds light on the role of time-parallel transport in eliminating—without approximation—all but an O(n) band of entries of the physical system’s force Jacobian. The result is a fast, unified, implicit treatment of viscous threads and elastic rods that closely reproduces a variety of fascinating physical phenomena, including hysteretic transitions between coiling regimes, competition between surface tension and gravity, and the first numerical fluid-mechanical sewing machine. The novel implicit treatment also yields an order of magnitude speedup in our elastic rod dynamics. [PDF] [bib] [Project] [Video]

Abstract:We develop an accurate, unified treatment of elastica. Following the method of resultant-based formulation to its logical extreme, we derive a higher-order integration rule, or elaston, measuring stretching, shearing, bending, and twisting along any axis. The theory and accompanying implementation do not distinguish between forms of different dimension (solids, shells, rods), nor between manifold regions and non-manifold junctions. Consequently, a single code accurately models a diverse range of elastoplastic behaviors, including buckling, writhing, cutting and merging. Emphasis on convergence to the continuum sets us apart from early unification efforts. [PDF] [bib] [Video]

Abstract:We analyze complex crack problems in elastic media using harmonic enrichment functions in a higher-order extended finite element implementation. The numerically computed enrichment function of a crack is the solution of the Laplace equation with discontinuous Dirichlet boundary condition along the crack, and its interaction with branches or other cracks is realized by imposing vanishing Neumann boundary conditions along those cracks. The classical finite element displacement approximation is enriched by adding the enrichment function of a crack through the framework of partition of unity. A nested subgrid mesh is used in the Laplace solve with a rasterized approximation of a crack, which simplifies the numerical integration—no partitioning of finite elements is required. Harmonic enrichment functions readily permit the extension to handle multiple interacting and branched cracks without any special treatment around the junction points. Several numerical examples are presented that affirm the accuracy and effectiveness of the method when applied to complex crack configurations under mixed-mode loading conditions. [PDF] [bib]

Abstract:A unifying procedure to numerically compute enrichment functions for elastic fracture problems with the extended finite element method is presented. Within each element that is intersected by a crack, the enrichment function for the crack is obtained via the solution of the Laplace equation with Dirichlet and vanishing Neumann boundary conditions. A single algorithm emanates for the enrichment field for multiple cracks as well as intersecting and branched cracks, without recourse to special cases, which provides flexibility over existing approaches in which each case is treated separately. Numerical integration is rendered to be simple—there is no need for partitioning of the finite elements into conforming subdivisions for the integration of discontinuous or weakly singular kernels. Stress intensity factor computations for different crack configurations are presented to demonstrate the accuracy and versatility of the proposed technique. [PDF] [bib]

Abstract:We develop a method for reliable simulation of elastica in complex contact scenarios. Our focus is on firmly establishing three parameter-independent guarantees: that simulations of well-posed problems (a) have no interpenetrations, (b) obey causality, momentum- and energy-conservation laws, and (c) complete in finite time. We achieve these guarantees through a novel synthesis of asynchronous variational integrators, kinetic data structures, and a discretization of the contact barrier potential by an infinite sum of nested quadratic potentials. In a series of two- and three dimensional examples, we illustrate that this method more easily handles challenging problems involving complex contact geometries, sharp features, and sliding during extremely tight contact. [PDF] [bib] [Project] [Video]

Abstract:Transferring existing mesh deformation from one character to another is a simple way to accelerate the laborious process of mesh animation. In many cases, it is useful to preserve the semantic characteristics of the motion instead of its literal deformation. For example, when applying the walking motion of a human to a flamingo, the knees should bend in the opposite direction. Semantic deformation transfer accomplishes this task with a shape space that enables interpolation and projection with standard linear algebra. Given several example mesh pairs, semantic deformation transfer infers a correspondence between the shape spaces of the two characters. This enables automatic transfer of new poses and animations. [PDF] [bib] [Video]

Abstract:We derive a variational integrator for certain highly oscillatory problems in mechanics. To do this, we take a new approach to the splitting of fast and slow potential forces: rather than splitting these forces at the level of the diﬀerential equations or the Hamiltonian, we split the two potentials with respect to the Lagrangian action integral. By using a diﬀerent quadrature rule
to approximate the contribution of each potential to the action, we arrive at a geometric integrator that is implicit in the fast force and explicit in the slow force. This can allow for signiﬁcantly longer time steps to be taken (compared to standard explicit methods, such as Störmer/Verlet) at the cost of only a linear solve rather than a full nonlinear solve. We also analyze the stability of this method, in particular proving that it eliminates the linear resonance instabilities that can arise with explicit multiple-time-stepping methods. Next, we perform some numerical experiments, studying the behavior of this integrator for two test problems: a system of coupled linear oscillators, for which we compare against the resonance behavior of the r-RESPA method; and slow energy exchange in the Fermi–Pasta–Ulam problem, which couples fast linear oscillators with slow nonlinear oscillators. Finally, we prove that this integrator accurately preserves the slow energy exchange between the fast oscillatory components, which explains the numerical behavior observed for the Fermi–Pasta–Ulam problem.
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Abstract:We present a discrete treatment of adapted framed curves, parallel transport, and holonomy, thus establishing the language for a discrete geometric model of thin flexible rods with arbitrary cross section and undeformed configuration. Our approach differs from existing simulation techniques in the graphics and mechanics literature both in the kinematic description — we represent the material frame by its angular deviation from the natural Bishop frame — as well as in the dynamical treatment — we treat the centerline as dynamic and the material frame as quasistatic. Additionally, we describe a manifold projection method for coupling rods to rigid-bodies and simultaneously enforcing rod inextensibility. The use of quasistatics and constraints provides an efficient treatment for stiff twisting and stretching modes; at the same time, we retain the dynamic bending of the centerline and accurately reproduce the coupling between bending and twisting modes. We validate the discrete rod model via quantitative buckling, stability, and coupled-mode experiments, and via qualitative knot-tying comparisons. [PDF] [bib] [Project] [Video]

Abstract:Example-based texture synthesis algorithms have gained widespread popularity for their ability to take a single input image and create a perceptually similar non-periodic texture. However, previous methods rely on single input exemplars that can capture only a limited band of spatial scales. For example, synthesizing a continent-like appearance at a variety of zoom levels would require an impractically high input resolution. In this paper, we develop a multiscale texture synthesis algorithm. We propose a novel example-based representation, which we call an exemplar graph, that simply requires a few low-resolution input exemplars at different scales. Moreover, by allowing loops in the graph, we can create inﬁnite zooms and inﬁnitely detailed textures that are impossible with current example-based methods. We also introduce a technique that ameliorates inconsistencies in the user’s input, and show that the application of this method yields improved interscale coherence and higher visual quality. We demonstrate optimizations for both CPU and GPU implementations of our method, and use them to produce animations with zooming and panning at multiple scales, as well as static gigapixel-sized images with features spanning many spatial scales. [PDF] [bib] [Project] [Video]

Abstract:Robust treatment of complex collisions is a challenging problem in cloth simulation. Some state of the art methods resolve collisions iteratively, invoking a fail-safe when a bound on iteration count is exceeded. The best-known fail-safe rigidifies the contact region, causing simulation artifacts. We present a fail-safe that cancels impact but not sliding motion, considerably reducing artificial dissipation. We equip the proposed fail-safe with an approximation of Coulomb friction, allowing finer control of sliding dissipation. [PDF] [bib] [Project] [Video]

Abstract:Discrete Laplace operators are ubiquitous in applications spanning geometric modeling to simulation. For robustness and efficiency, many applications require discrete operators that retain key structural properties inherent to the continuous setting. Building on the smooth setting, we present a set of natural properties for discrete Laplace operators for triangular surface meshes. We prove an important theoretical limitation: discrete Laplacians cannot satisfy all natural properties; retroactively, this explains the diversity of existing discrete Laplace operators. Finally, we present a family of operators that includes and extends well-known and widely-used operators. [PDF] [bib]

Abstract:We propose a new method named compressive structured light for recovering inhomogeneous participating media. Whereas conventional structured light methods emit coded light patterns onto the surface of an opaque object to establish correspondence for triangulation, compressive structured light projects patterns into a volume of participating medium to produce images which are integral measurements of the volume density along the line of sight. For a typical participating medium encountered in the real world, the integral nature of the acquired images enables the use of compressive sensing techniques that can recover the entire volume density from only a few measurements. This makes the acquisition process more efficient and enables reconstruction of dynamic volumetric phenomena. Moreover, our method requires the projection of multiplexed coded illumination, which has the added advantage of increasing the signal-to-noise ratio of the acquisition. Finally, we propose an iterative algorithm to correct for the attenuation of the participating medium during the reconstruction process. We show the eﬀectiveness of our method with simulations as well as experiments on the volumetric recovery of multiple translucent layers, 3D point clouds etched in glass, and the dynamic process of milk drops dissolving in water.
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Abstract:We combine the often opposing forces of artistic freedom and mathematical
determinism to enrich a given animation or simulation of a surface with physically based detail. We present a process called tracking, which takes as input a rough animation or simulation and enhances it with physically simulated detail. Building on the foundation of constrained Lagrangian mechanics, we propose weak-form constraints for tracking the input motion. This method allows the artist to choose where to add details such as characteristic wrinkles and folds of various thin shell materials and dynamical effects of physical forces. We demonstrate multiple applications
ranging from enhancing an artist’s animated character to guiding a simulated inanimate object. [PDF] [bib] [Project] [Video]

Abstract:Many textiles do not noticeably stretch under their own weight. Unfortunately, for better performance many cloth solvers disregard this fact. We propose a method to obtain very low strain along the warp and weft direction using Constrained Lagrangian Mechanics and a novel fast projection method. The resulting algorithm acts as a velocity filter that easily integrates into existing simulation code. [PDF] [bib] [Project] [Video]

Abstract:Filtering is critical for representing image-based detail, such as textures or normal maps, across a variety of scales. While mipmapping textures is commonplace, accurate normal map filtering remains a challenging problem because of nonlinearities in shading¡ªwe cannot simply average nearby surface normals. In this paper, we show analytically that normal map filtering can be formalized as a spherical convolution of the normal distribution function (NDF) and the BRDF, for a large class of common BRDFs such as Lambertian, microfacet and factored measurements. This theoretical result explains many previous filtering techniques as special cases, and leads to a generalization to a broader class of measured and analytic BRDFs. Our practical algorithms leverage a significant body of previous work that has studied lighting-BRDF convolution. We show how spherical harmonics can be used to filter the NDF for Lambertian and low-frequency specular BRDFs, while spherical von Mises-Fisher distributions can be used for high-frequency materials. [PDF] [bib] [Project] [Video]

Abstract:Hinge-based bending models are widely used in the physically-based animation of cloth, thin plates and shells. We propose a hinge-based model that is simpler to implement, more efficient to compute, and offers a greater number of effective material parameters than existing models. Our formulation builds on two mathematical observations: (a) the bending energy of curved flexible surfaces can be expressed as a cubic polynomial if the surface does not stretch; (b) a general class of anisotropic materials—those that are orthotropic—is captured by appropriate choice of a single stiffness per hinge. Our contribution impacts a general range of surface animation applications, from isotropic cloth and thin plates to orthotropic fracturing thin shells. [PDF] [bib] [Video]

Abstract:[This is preprint--not a final copy.] We present a family of discrete isometric bending models (IBMs) for triangulated surfaces in 3-space. These models are derived from an axiomatic treatment of discrete Laplace operators, using these operators to obtain linear models for discrete mean curvature from which bending energies are assembled. Under the assumption of isometric surface deformations we show that these energies are quadratic in surface positions. The corresponding linear energy gradients and constant energy Hessians constitute an efficient model for computing bending forces and their derivatives, enabling fast time-integration of cloth dynamics with a two- to three-fold net speedup over existing nonlinear methods, and near-interactive rates for Willmore smoothing of large meshes.
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Abstract:Discrete curvature and shape operators, which capture complete information about directional curvatures at a point, are essential in a variety of applications: simulation of deformable two-dimensional objects, variational modeling and geometric data processing. In many of these applications, objects are represented by meshes. Currently, a spectrum of approaches for formulating curvature operators for meshes exists, ranging from highly accurate but computationally expensive methods used in engineering applications to efficient but less accurate techniques popular in simulation for computer graphics. We propose a simple and efficient formulation for the shape operator for variational problems on general meshes, using degrees of freedom associated with normals. On the one hand, it is similar in its simplicity to some of the discrete curvature operators commonly used in graphics; on the other hand, it passes a number of important convergence tests and produces consistent results for different types of meshes and mesh refinement.
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A note on the triangle-centered quadratic interpolation discretization of the shape operator Jason Reisman, Eitan Grinspun, Denis Zorin, Technical Report, Department of Computer Science, Columbia University and New York University, 2007.

Abstract:In this note we consider a simple shape operator discretization for general meshes, based on computing an interpolating quadratic function passing through vertices of a triangle and its edge-adjacent neighbors. This approximation is computationally simple and consistent for a broad class of meshes. However, its convergence properties in the context of mesh optimization problems are not as good as some of the previously proposed techniques and it suffers from instabilities for certain point configurations.
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Abstract:Advances in high-performance computing have led to the broad use of computational studies in everyday engineering and scientific applications. A single study may require thousands of computational experiments, each corresponding to individual runs of simulation software with different parameter settings; in complex studies, the pattern of parameter changes is complex and may have to be adjusted by the user based on partial simulation results. Unfortunately, existing tools have limited high-level support for managing large ensembles of simultaneous computational experiments. In this paper, we present a system architecture for interactive computational studies targeting two goals. The first is to provide a framework for high-level user interaction with computational studies, rather than individual experiments; the second is to maximize the size of the studies that can be performed at close to interactive rates. We describe a prototype implementation of the system and demonstrate performance improvements obtained using our approach for a simple model problem. [PDF] [bib]

Abstract:Efficient computation of curvature-based energies is important for practical implementations of geometric modeling and physical simulation applications. Building on a simple geometric observation, we provide a version of a curvature-based energy expressed in terms of the Laplace operator acting on the embedding of the surface. The corresponding energy, being quadratic in positions, gives rise to a constant Hessian in the context of isometric deformations. The resulting isometric bending model is shown to significantly speed up common cloth solvers, and when applied to geometric modeling situations built on Willmore flow to provide runtimes which are close to interactive rates. [PDF] [bib] [Project] [Video]

Abstract:We introduce a way of simulating the creation of simple Origami (paper folding). The Origami is created in a thin shell simulation that realistically models the behavior and physical properties of paper. We demonstrate how to fold and crease the simulated paper wherever the user desires. [PDF] [bib]

Abstract:Current PRT methods exploit spatial coherence of the lighting (such as with wavelets) and of light transport (such as with CPCA). We consider a significant, yet unexplored form of coherence, temporal coherence of the lighting from frame to frame. We achieve speedups of 3x-4x over conventional PRT with minimal implementation effort, and can trivially be added to almost any existing PRT algorithm. [PDF] [bib] [Project] [Video]

Abstract:We introduce a method for simulating the inelastic deformation of thin shells: we model plasticity and fracture of curved, deformable objects such as light bulbs, egg-shells and bowls. Our novel approach uses triangle meshes yet evolves fracture lines unrestricted to mesh edges. We present a novel measure of bending strain expressed in terms of surface invariants such as lengths and angles. We also demonstrate simple techniques to improve the robustness of standard timestepping as well as collision response algorithms. [PDF] [bib] [Project]

Abstract:In this paper we introduce a discrete shell model describing the behavior of thin flexible structures, such as hats, leaves, and aluminum cans, which are characterized by a curved undeformed configuration. Previously such models required complex continuum mechanics formulations and correspondingly complex algorithms. We show that a simple shell model can be derived geometrically for triangle meshes and implemented quickly by modifying a standard cloth simulator. Our technique convincingly simulates a variety of curved objects with materials ranging from paper to metal, as we demonstrate with several examples including a comparison of a real and simulated falling hat [PDF] [bib]

Abstract:Finite element solvers are a basic component of simulation applications; they are common in computer graphics, engineering, and medical simulations. Although adaptive solvers can be of great value in reducing the often high computational cost of simulations they are not employed broadly. Indeed, building adaptive solvers can be a daunting task especially for 3D finite elements. In this paper we are introducing a new approach to produce conforming, hierarchical, adaptive refinement methods (CHARMS). The basic principle of our approach is to refine basis functions, not elements. This removes a number of implementation headaches associated with other approaches and is a general technique independent of domain dimension (here 2D and 3D), element type (eg, triangle, quad, tetrahedron, hexahedron), and basis function order (piecewise linear, higher order B-splines, Loop subdivision, etc.). The (un-)refinement algorithms are simple and require little in terms of data structure support. We demonstrate the versatility of our new approach through 2D and 3D examples, including medical applications and thin-shell animations. [PDF] [bib]

Abstract:Current formulations of adaptive finite element mesh refinement seem simple enough, but their implementations prove to be a formidable task. We offer an alternative approach called CHARMS: Conforming Hierarchical Adaptive Refinement Methods. Our method yields equivalent adapted approximation spaces wherever the traditional mesh refinement is applicable, but proves to be significantly simpler to implement. At the same time it is much more powerful in that it is general (no special tricks are required for different types of finite elements), and applicable for some newer approximations where traditional mesh refinement concepts are not of much help, for instance on subdivision surfaces. [PDF] [bib]

Abstract:Interference detection is vital for simulation and animation. Our interest was born of a larger project: using the Subdivision Element Method and the thin-shell equations we produce realistic animations of crushing, crumpling, and wrinkling. In this paper we derive normal bounds for subdivision surfaces and use these to develop an efficient algorithm for collision detection with specific optimizations for self-interference. The normal bounds are also useful for CAD and rendering [PDF] [bib]

Abstract:Numerically accurate simulation of the mechanical behavior of thin flexible structures is important in application areas ranging from engineering design to animation special effects. Subdivision surfaces provide a unique opportunity to integrate geometric modeling with concurrent finite element analysis of thin flexible structures. Their mechanics are governed by the so-called thin-shell equations. We present a concise treatment of thin-shell equations including dynamic behavior, scalable material models, and the treatment of collisions (detection as well as response). The resulting energy minimization problem is non-linear and in turn able to capture effects of far more realism than linear models. We demonstrate these claims with a number of simulations which exhibit characteristic effects of real world experiments [PDF] [bib]

The Basis Refinement MethodEitan Grinspun, PhD Thesis, Caltech, 2003.

Abstract:Finite element solvers are critical in computer graphics, engineering, medical and biological application areas. For large problems, the use of adaptive refinement methods can tame otherwise intractable computational costs. Current formulations of adaptive finite element mesh refinement seem straightforward, but their implementations prove to be a formidable task. We offer an alternative point of departure which yields equivalent adapted approximation spaces wherever the traditional mesh refinement is applicable, but proves to be significantly simpler to implement. At the same time it is much more powerful in that it is general (no special tricks are required for different types of finite elements), and applicable to novel discretizations where traditional mesh refinement concepts are not of much help, for instance on subdivision surfaces. [PDF] [bib]